Compensation of Nonlinearities in Transducers

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Compensation of Nonlinearities in Transducers

Martin Rune Andersen

Kgs. Lyngby 2005

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Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800 Lyngby, Denmark Phone +45 45253351, Fax +45 45882673 reception@imm.dtu.dk

www.imm.dtu.dk

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Abstract

This thesis is concerned with the topic of compensation algorithms for the nonlinearities in transducers for use in hifi-loudspeaker. The thesis contains a examination of the general ideal loudspeaker model.

Furthermore, nonlinearities that influences the performance of the loudspeaker are described, and measurement are done on what they does to the loudspeaker.

The loudspeaker model is from here transformed into the digital domain, in preparation for constructing a control system later. In order to make this model complete, appropriate functions are fitted to the nonlinearities and added to the model.

Furthermore, a text based toolbox for Matlab is made for simulations and evaluations of different properties in both the loudspeaker and in the upcoming compensation algorithm.

Finally, two interesting feedforward controller systems are presented. The first is the ”state space” compensator and the second is the ”Mirror filter” derived by Wolfgang Klippel. Later research has shown that they are of same controller type.

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Resum´ e

Denne rapport omhandler emnet kompenserings algoritmer for ikke-lineariteter i transducere til brug i hi-fi højttalere. Rapporten indeholder en gennemgang af den generelle ideelle højttaler model. Yderligere er ikke-lineariteterne der forringer ydelsen af højttaleren beskrevet og m˚alinger, omkring hvad det gør ved denne, er foretaget.

Herfra overføres højttaler modellen til det digitale domæne med henblik p˚a senere at konstruere et regu- lering system til denne. Og for at gøre modellen komplet i forhold til højttaleren, tilpasses en passende funktion til ikke-lineariteterne s˚a disse kan inkluderes.

Desuden præsenteres en Matlab toolbox til hjælp med at simulere og evaluere de forskellige egenskaber i b˚ade højttalere og ogs˚a i de senere kompenseringsalgoritmer.

Til sidst gennemg˚as to interessante prekompensering algoritmer. Den ene er ”state-space” kompensatoren og den anden er Wolfgang Klippels ”Mirror filter”. Ved senere tids forskning har det dog vist sig, at disse to er af samme type reguleringssystem.

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Preface

This thesis was written to fulfil the requirements to obtain a master of science degree in engineering from the Technical University of Denmark.

It was carried out in corporation with Oticon A/S, at IMM-DTU and Oersted-DTU. Furthermore, i would like to thank Bang & Olufsen ICEpower A/S for supplying me with articles from audio engineer society (AES), and giving me the opportunity for a one day trip to Bang & Olufsen in Struer.

I would like to thank my supervisors Ole Winther, Jan Larsen, Finn Agerkvist and Steen M. Munk for giving my constructive feedback, which helped me in writing my thesis.

I would like thank my girlfriend Jeanette for her support during my project and for her house keeping in the end of my project while I was stocked with my project at DTU (I hope she will continue with that :)).

I would like to thank Fares El Azm and Crilles Bak Rasmussen for relevant technical discussions and for proofreading my thesis. Finally, I would like to thank some of my friends, Martin, Denis, Maya, Robert, Fares and Daniel, for a good atmosphere while writing our thesis.

Martin Rune Andersen, 25th May 2005, Lyngby

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List of Figures

1.1 Commercial surround systems with wireless speakers (a) Sony HT SL900, (b) Pioneer DCS

515 . . . 2

2.1 Typical transducer construction . . . 6

2.2 (a) Overhung voice-coil, (b) Underhung voice-coil . . . 6

2.3 Schematic of electric circuit of an transducer . . . 7

2.4 SDOF mechanical diagram for the transducer . . . 8

2.5 Speaker mounted in a closed-box enclosure (cavity) . . . 10

2.6 Speaker mounted in a vented-box enclosure . . . 11

2.7 Acoustic radiation impedance and approximations to it . . . 13

2.8 SPL at 1m for a closed box loudspeaker . . . 14

2.9 Displacement for a closed box loudspeaker, 0dB = 1mm . . . 15

2.10 SPL at 1m for a vented box loudspeaker . . . 16

2.11 Displacement for a vented box loudspeaker, 0dB = 1mm . . . 16

2.12 Linear thermal model . . . 17

2.13 Circuit of electrical part with eddy current loss . . . 18

2.14 measured impedance and fitted impedance, (a) s1, (b) s2 . . . 20

2.15 Drawings of the loudspeaker box . . . 23

2.16 Measurements of box alignment, (a) corrected sound pressure insight of box, (b) electrical impedance . . . 25

3.1 Nonlinear force factorB(xD)l, and mirrored to get a better asymmetry view . . . 28

3.2 Nonlinear inductanceLE(xD) of the voice-coil . . . 28

3.3 Nonlinear complianceCD(xD), and mirrored to get a better asymmetry view . . . 29

3.4 Cross section of typical single roll suspension . . . 30

3.5 Deviation of the sound pressure compared to the inverted impedance . . . 31

3.6 (a) Resonance frequencyfsand (b) quality factorQT S, at different displacements . . . . 34

3.7 Normalized SPL at low frequencies, measured inside box . . . 35

3.8 DC in diaphragm displacement at different driving levels . . . 35

3.9 (a) Total harmonic distortion (b) Intermodulation distortion . . . 36

3.10 Simulated total harmonic distortion, onlyBlnonlinearity . . . 37

3.11 Simulated total harmonic distortion, onlyCDnonlinearity . . . 37

3.12 Simulated total harmonic distortion, onlyLE nonlinearity . . . 37

3.13 Simulated intermodulation distortion, onlyBlnonlinearity . . . 38

3.14 Simulated intermodulation distortion, onlyCD nonlinearity . . . 38

3.15 Simulated intermodulation distortion, onlyLE nonlinearity . . . 38

4.1 Fit on data . . . 41

4.2 Polynomial fit on force factor data data . . . 41

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4.5 Sum of squares error . . . 44

4.6 Bad fits, (a)σl= 1, (b)σl= 4.5 . . . 44

4.7 σ= 2.81, range of centers{−7mm; 7mm}, no offset . . . 45

4.8 σ= 2.55, range of centers{−7mm; 7mm}, no offset . . . 45

4.9 σ= 3, range of centers {−6mm; 6mm}, offset equal toBl(0) . . . 45

4.10 Fit on inductance data with sigmoid function, (a) iterations in fit, (b) final result . . . 47

4.11 Polynomial fit on compliance data, where only offset is changed . . . 48

5.1 Transfer function of discrete derivatives,f_s= 48kHz, (a) magnitude, (b) phase . . . 51

5.2 Block diagram of loudspeaker compensation toolbox in Matlab . . . 57

5.3 THD simulation plotted along with belonging measurements . . . 59

5.4 IMD simulation plotted along with belonging measurements . . . 60

6.1 Three types of controller systems, (a) feed-forward, (b) feedback, (c) adaptive feed-forward 62 6.2 State-space compensator . . . 65

6.3 State-space compensator with state observer . . . 68

6.4 State-space compensator with pre-simulated states . . . 68

6.5 THD and IMD simulations at different driving levels, (a) THD, (b) IMD . . . 69

6.6 THD and IMD simulations at different driving levels, (a) THD, (b) IMD . . . 71

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List of Tables

2.1 Linear parameters for the drivers used in the test loudspeaker . . . 21 2.2 Calculated box parameters in alignment . . . 22 2.3 Measured box parameters in alignment . . . 24

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Abbreviation

B&K Br¨uel & Kjear

DST Danish Sound Technology EMF Elector magnetic force IMD InterModulation distortion PA Public address

SDOF Single-degree of freedom THD Total Harmonic Distortion

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Notation

aD Acceleration of diaphragm [m/s²] B Magnetic flux density [T]

c Speed of sound [343 m/s]

C_D Mechanical compliance of diaphragm suspension [m/N]

C_t Mechanical compliance of diaphragm suspension

and mechanical equivalent of acoustic compliance [m/N]

C_tb Thermal capacity of box Ctm Thermal capacity of magnet Ctv Thermal capacity of voice-coil

f Frequency [1/s]

fB Helmholtz resonance frequency [1/s]

FD Force on voice-coil [N]

FMD Force produced from the mass of diaphragm when moving [N]

FRD Force produced from the resistance of diaphragm when moving [N]

FCD Force produced from the compliance of diaphragm when moving [N]

ic Current flow in voice-coil [A]

l Effective length of voice-coil wire in the magnetic field [m]

K_D Mechanical stiffness of diaphragm suspension [N/m]

L₂ Para-inductance of voice-coil [H]

L_E Inductance of voice-coil [H]

L_P Length of vent [m]

L_{P,ef f} Effective length of vent [m]

MA1 Acoustical mass in front of the diaphragm MAB Acoustical mass behind of the diaphragm MD Mass of diaphragm and coil assembly [kg]

Mt Mass of diaphragm and coil assembly and mechanical equivalent of acoustic mass [kg]

Pt Input power in thermal model [W]

Q0 Total quality factor of driver in vented box QT C Total quality factor of driver in closed box QT S Total quality factor of driver in free air QL Quality factor of loss

QE Electrical quality factor of driver in free air Q_M Mechanical quality factor of driver in free air

p_D Pressure difference between front and back of diaphragm [Pa]

p_c Pressure inside box [Pa]

R₂ Electrical resistance due to eddy current losses R_E Resistance in coil wire [Ω]

RD Mechanical resistance of diaphragm suspension [N·s/m]

Rt Mechanical resistance of diaphragm and coil assembly and mechanical equivalent of acoustic resistance [N·s/m]

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R_tb Thermal resistance of box R_tm Thermal resistance of magnet R_tv Thermal resistance of voice-coil

s Laplace frequency, notation for−jω in phasor equations [s⁻¹] S_B Inside cross area of front plate [m²]

SD Cross area of diaphragm [m²] SP Cross area of vent [m²] Ta Ambient temperature [^◦C]

Tb Box temperature [^◦C]

Tm Temperature of magnet [^◦C]

Tv Temperature of voice-coil [^◦C]

u Voltage applied on the terminals [V]

uD Mechanical velocity of the diaphragm [m/s]

UB Volume velocity emitted into the box U_D Volume velocity emitted by the diaphragm U_L Volume velocity emitted by air leaks

U₀ Total volume velocity emitted by loudspeaker U_P Volume velocity emitted by vent

v Voltage applied to the compensation system [V]

vc Voltage drop across the terminals of the voice-coil [V]

Vc Volume of box [l]

VAB Effective volume of box [l]

w Voltage applied to the linear dynamics [V]

xD Diaphragm displacement [m]

Zrad_b Acoustic radiation impedance in back of the diaphragm Zrad_F Acoustic radiation impedance in front of the diaphragm Zrad Acoustic radiation impedance

ZE Impedance of voice-coil [Ω]

ZEB Impedance of blocked voice-coil (no diaphragm movements) [Ω]

Z_M Mechanical impedance [Ω]

Z_rm Mechanical equivalent of acoustical impedance

∆T_c Increase of cabinet temperature [^◦C]

∆T_m Increase of magnet temperature [^◦C]

∆Tmss Steady-state magnet temperature [^◦C]

∆Tv Increase of voice-coil temperature [^◦C]

∆Tvss Steady-state voice-coil temperature [^◦C]

ρ0 Density of air 1.18 at 20^◦C ρ Effective density

λ Wavelength of soundc/f ω Angular frequency 2πf [rad/s]

ωs Undamped natural frequency driver in free air [s⁻¹] ωs Natural frequency driver in closed box [s⁻¹]

ξ Damping ratio of the mechanical system without acoustical loading

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Chapter 1 Introduction

There exists clear division between people who are willing to pay a lot for hifi-systems and people who are not.

The group which are willing to pay more do not compromise the performance in terms of the price, and tend to spend a fortune on components which only slightly increase the performance. On the other hand the second group consider the price more important than the performance.

The intension for doing this project was to challenge both of the two worlds by looking at the perspective from a different point of view. This would be done by taking a cheap hifi-system, and then digitally compensating for the distortion, only modestly increasing the price to the extent that the last of the two groups would accept the increase in the price.

During thesis a symposium, that took place the December 2. at IMM-DTU in corporation with Oersted- DTU, was arranged, where two experts within transducer linearization, Dr. Wolfgang Klippel and PhD.

Andrew Bright from Nokia, were invited. The symposium ended up in a panel discussion, where also some among the industry joined. The topic was ”how far are we from commercial products?”. The overall conclusion of the discussion was that in order to see commercial success, the implementation must be done in a application where a DSP-processor already is available. Furthermore, the compensation algorithm must be less complex than the rest of the DSP code.

In figure 1.1 two products, from respectively Sony and Pioneer, that are on the market are shown. The interesting with these systems are that they include wireless speakers, which then leads to active speakers with digital inputs. So the speaker in these systems already contain some kind of digital processing.

Furthermore, the amplifier in the Sony system is their digital one with S-master technology, which makes the electronics in the Sony speaker completely digital.

Of other applications of interest to this thesis are mobile units in general, as mobile phones, hand held computers and hearing aids.

1.1 History

The moving-coil transducer for audio reproduction was first described in 1874 by Ernst Werner Siemens.

But not until 1925 the transducer appeared as a loudspeaker as we know it today, developed by Chester W. Rice and Edward W. Kellogg.

From the beginning research, on compensating the nonlinearities in a transducer, has been done, but not until the last decade useable proposals have been made and furthermore, not until the last few years has the proposals been realizable.

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Figure 1.1: Commercial surround systems with wireless speakers (a) Sony HT SL900, (b) Pioneer DCS 515

1.2 Purpose of the thesis

The purpose of this thesis is to look into the theory of loudspeaker compensation and how to apply it on existing loudspeakers, thus optimization of the loudspeaker design is not considered in here. As many attempts have been done in the past, these must be studied and a short overview will be given.

The following short list describes the topics that need to be investigated:

• Transducers theory, how they are used in a loudspeaker and how they are modeled.

• Nonlinearities that must be added to the model, in order to describe it at large signals

• Attempts in the past to make control systems

• How to use loudspeaker compensation in applications

Furthermore, one or two algorithms must be investigated even further and be implemented in Matlab with real transducer data. Of this reason a simple test loudspeaker must be constructed to achieve these data.

1.3 Organization of the thesis

Chapter 1. Introduction

Chapter 2. Linear transducer model: Classical linear models of the loudspeaker are reviewed.

Furthermore, a test loudspeaker is constructed, where its parameters are used in later simulations.

Chapter 3. Nonlinearities in the transducer model: Nonlinearities in the transducer model are discussed, and it is finally decided which ones of biggest importance. Furthermore, distortion in the test loudspeaker is measured.

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Chapter 4. Modeling of nonlinearities: This chapter focusses on fitting models to the nonlinearities.

Different functions are applied and their advantages and disadvantages are discussed. Finally an idea for adding soft clipping to the transducer is presented.

Chapter 5. Discrete nonlinear simulation: In this chapter the transducer model is transformed into discrete time difference equations. From these general digital filters are derived, both linear and nonlinear filters. Finally a Matlab toolbox for simulating loudspeakers and compensation systems, is presented.

Chapter 6. Control: Some general control theory are reviewed. Later two feedforward compensator algorithms are presented and they are evaluated on the discrete nonlinear model.

Chapter 7. Future work: Based on the work done in this thesis, a schedule for how to carry on, is given.

Chapter 8. Conclusions: Conclusions are drawn on active control of loudspeakers using feedforward processing.

Appendix A. Data coefficients: Filter coefficients and other data are given in here.

Appendix B. Parameters of the test loudspeaker: Parameters for the drivers used in the loudspeaker and the loudspeaker coefficient, are given.

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Chapter 2 Linear transducer model

In the this chapter the basic linear transducer model is described. The model, which can be written as basic filters, was completed by Thiele [Thiele, 1961] and in a series of papers by Small [Small, 1971], [Small, 1972], [Small, 1973a], [Small, 1973b], [Small, 1973c], [Small, 1973d] and [Small, 1973e]. Though Thiele and Small’s papers gives a full description of the model, [Leach, 1999] and [Bright, 2002] have been a great inspiration for the study of the linear transducer model.

At the end of this chapter, a vented-box loudspeaker is constructed. The parameters of the transducer used in this, is used in Matlab simulations throughout this thesis and the loudspeaker is used for testing the derived control systems.

2.1 Transducer construction

A typical construction of a moving-coil transducer is seen at figure 2.1. The transducer consists of three circuits:

• The electrical circuit, which is a voice-coil and the resistance in the coil wire.

• The mechanical circuit, which is a single-degree-of-freedom (SDOF) mechanical oscillator.

• The acoustical circuit, which is the air loading both in front and in back of the diaphragm.

The sound is radiated by the diaphragm, which is the moving surface. The diaphragm is moved by the voice-coil, which is a part of the electromagnetic network. The other part of the electromagnetic network is the magnet and its circuit (pole pieces). If current is applied to the voice-coil in the magnetic field, a force will move the voice-coil. If the current is reversed, the force is reversed too. The force on the diaphragm is linearly related to the current in the voice-coil if the number of turns of wire in the magnetic field does not change when the diaphragm moves. In figure 2.2, two methods of how this is achieved, is seen. If the coil is overhung as in figure 2.2a, then as the coil moves in one direction, some turns leave the gap while others enter it in the other end and the total number of turns of wire is kept constant. In 2.2b the underhung method is seen, when the coil moves no turns of wire leave the gap and non enter, thus keeping the number of turns constant.

If the transducer is pushed too far, the coil moves out of the gap decreasing the number of turns of wire, and it becomes nonlinear.

An air vent is often used to prevent the air in the small cavity behind the dust cap from being nonlinear and furthermore it increases the air convection which gives a much better cooling of the coil, see [Klippel, 2003].

The diaphragm is kept in place by the spider and the outer suspension. As here, two suspensions is often used, but in micro-transducers only the outer suspension is used.

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Figure 2.1: Typical transducer construction

Figure 2.2: (a) Overhung voice-coil, (b) Underhung voice-coil

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Figure 2.3: Schematic of electric circuit of an transducer

2.2 Electrical circuit

The electrical circuit consists of the voice-coil, a electrical resistance in the wire and voltage applied on the coil when moving in a magnet field. The schematic of the electrical circuit is seen in figure 2.3. uis the input voltage and vc is the voltage on the voice-coil when moving in a magnet field, also called the back electro magnetic force (back EMF). The value of the voltage is proportional to the velocity of the voice-coil and thus becomes as a controlled voltage source in the electrical circuit. REis the resistance in the coil wire,LE is the inductance and ic is the current in the voice-coil. From this the voltage equation can be written as:

u(t) =REic(t) +LE

di_c(t)

dt +vc(t) (2.1)

All parameters in the equation are constant and it is therefore solved easily with the Laplace transform, and thus the voltage equation can be written as:

u(s) =R_Ei_c(s) +L_Esi_c+v_c(s) (2.2) wheresis the ’Laplace variable’ equal to−jω,ω= 2πf andfis the frequency hertz. And finally, grouping RE and LE to a electrical impedance ZEB, which is the blocked electrical impedance as the electrical impedance, caused by the movements of the diaphragm (see section 2.4), is left out:

u(s) =ZEB(s)ic+vc(s) (2.3)

2.3 Mechanical circuit

The mechanical SDOF network is the diaphragm with coil assembly and the two suspensions. A diagram of this is seen in figure 2.4. MD is the mass of the diaphragm and coil assembly, RD is the loss in the diaphragm suspension, CD is the compliance of the suspension and KD = 1/CD is the stiffness of the suspension. Zrm is the mechanical-equivalent of the acoustical radiation impedance and is dealt with in the next section. Each of these three components produce a force when the diaphragm is moving:

F_{M D}(t) =M_Dd²xD(t)

dt² F_RD(t) =R_DdxD(t)

dt F_CD(t) = 1 CD

x_D(t) (2.4)

The first of the equations is Newton’s second law, where the second order derivative of the diaphragms displacement is the acceleration. The second equation is the force occurring when a object is moving with a certain speed and a resistance is applied on it; where the first order derivative of the diaphragms displacement is the velocity of it. The force has the opposite direction as the velocity, stopping the diaphragm. The last equation is the spring equation. In this, the direction of the force is always towards the rest position of the diaphragm.

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Figure 2.4: SDOF mechanical diagram for the transducer

Hereof the system can be described by the second-order inhomogeneous differential equation:

FD(t) =MDx¨D(t) +RDx˙D(t) + 1 CD

xD(t) +ZrmxD(t) (2.5) whereFD(t) is the force applied on the diaphragm and coil assembly when current flows in the coil. Taking the Laplace transformation of it gives:

FD(s) = (MDs²+RDs+ 1

C_D +Zrm)xD(s) (2.6)

If the acoustical loading is left out, the mechanical impedance can be found:

ZM0(s) = F_D(s) uD(s) ic(s)=0

pD(s)=0

=MDs+RD+ 1

CDs (2.7)

where uD(s) = ˙xD(s). Here the voice-coil is open circuited and there is no acoustical loading (pressure difference from front to back of the diaphragm,pD, is zero).

With the electrical and mechanical circuit, the resonance frequency and total quality factor for the driver in free air, is given by:

ωs= 2πfs= 1

√MtCD

(2.8) Q_{T S}= 1

Bl²/RE+RD

rM_t CD

(2.9) M_t is the mass of the diaphragm and mechanical equivalent of the acoustical loading given in section 2.5.1. The combination of the electrical and mechanical circuit is described in the following section.

2.4 Electro-mechanical transduction

The classical electrodynamic interaction of the line currents and static magnetic field causes a force applied on the voice-coil. It is given by the effective length of the voice-coil wire in the magnetic field times the magnetic force factor times the current in the electrical circuit:

FD(t) =Blic(t) (2.10)

This is also called the electro magnetic force (EMF). The two first products on the right hand side (RHS) is referred to as theB·l product.

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The back EMF as given in the last term on the RHS in 2.1 applies a voltage on the voice-coil:

v_c(t) =Blu_D(t) (2.11)

This shows that the speaker is a regressive network.

2.5 Acoustical circuit

The acoustical circuit is the load on the loudspeaker, both in back of the diaphragmZ_rad_b and in front of it Zrad_f. The load in front is the room and the load in back of the diaphragm is a fixed box. The most commonly used types of rear loads is:

• Closed box.

• Vented box.

• Transmission line speaker.

• Dipole speaker.

• Infinite baffle.

• Horn loaded speaker.

The two first are the most commonly used constructions for amateur hifi and surround systems. The remaining are more seldom and are often used by enthusiastic persons or, for public address (PA) equip- ment. As this thesis aims at the cheap part of the amateur market, the first two are the ones dealt with in this thesis.

The closed box loudspeaker has a certain volume and separates the front from the back of the diaphragm.

This is the most simple type of loudspeaker existing.

The vented box loudspeaker is a closed box loudspeaker with a vent and which forms a resonating circuit.

The advantage of vented box loudspeakers is, if properly designed, the lower cut off frequency would be lower than with a properly designed closed box. The volume of the vented box would be bigger than it would for the closed box, though the efficiency is the same.

2.5.1 Closed box

Figure 2.5 is a closed box speaker system. The box has the volumeVC and the internal pressurepC(t). If the wavelength is about five times greater than any of the box dimensions,λ >∼5·(h, w, d), the pressure is assumed to be constant throughout the cavity; the wavelength is found by λ = c/f, where c is the velocity of sound andf is the frequency. SD is the cross area of the diaphragm.

The pressure difference between the front and the back of the diaphragm is given by:

pD(t) =UD(t)

MABs+RAB+ 1 CABs

+UD(t)(MA1s) (2.12)

where U_D = S_Du_D is the volume velocity radiated by the diaphragm, M_AB is the mass behind the diaphragm, M_A1 is the mass seen in front of the it, R_AB is the resistance in the box and C_AB is the compliance of the box. The acoustical radiation impedance for both the front and the back is written as:

Zrad_b=MABs+RAB+ 1

CABs Zrad_f =MA1 (2.13)

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Figure 2.5: Speaker mounted in a closed-box enclosure (cavity)

Only the mass and compliance inZrad_bcan be calculated, the resistance must be measured as it depends on how much filling there is put into the box. The mass and compliance is calculated as:

MAB= Bρ

πa [kg/m⁴] CAB =VAB

ρ₀c² [m⁵/N] (2.14)

where VAB is effective acoustic volume,ρ0 is the density of air,ρis effective density of the combined air and fiber filling in the box,Bis the mass loading factor andais the radius of the diaphragm. The effective volume (volume seen by the loudspeaker) is increased when filled with uncompressed fiber material. This is due to the fact that the filling fibers do not necessarily move with the same velocity as the air particles and furthermore, the specific heat is different for the two. Sometimes it is assumed that the increase of volume, when a closed box is used, is about 25%. With a vented box system the increase is less because normally only the inside walls are covered with filling.

The mass loading factor given in [Leach, 1999] is defined by the dimension of the box:

B=d√ π 3 ·S_D^3/2

S_B² + 8 3π

1−SD

S_B

(2.15) wheredis the depth of the box andS_B is the inside area of the wall in which the loudspeaker is mounted.

As mentioned the resistanceR_AB can not be calculated, thus it needs to be measured. The measurement is not done in this thesis

The acoustical loading in front of the diaphragm is, despite the simple mass model MA1, very complex.

The impedance can be well approximated with a mass, but the mass depends on the dimensions of the box or baffle it is mounted in. If the wavelength of sound is short (high frequencies) compared to the dimensions of the baffle, then the mass is equal to the one when mounted in a infinite baffle. But at very low frequencies where the wavelength is very long compared to the dimensions of the baffle, then the mass of a point source might be used; the mass when mounted in a infinite baffle is two times bigger than the mass seen by a point source. For both these cases, it is assumed that the sound is radiated as simple spherical waves; for the infinite baffle in 2πspace and for the point source in 4πspace.

In [Leach, 1999]¹ a third option is given, which is an alternative between the two. Here the driver is mounted in the end of a long tube where the sound waves diffracts into the space behind the box, and now simple spherical waves cannot be assumed. This model is considered to be the most optimal if the speaker is placed away from the wall, else the infinite baffle will suite best as the wall can be considered as a extension to the front of the speaker.

At high frequencies the infinite baffle would always be the optimal choice, but since the displacement behaves as a low-pass filter, as shown later, the greatest displacements are exhibited at low frequencies, thus the model that describes the low frequencies best must be chosen. Therefore the last described model is used in this thesis and it is given by:

MA1=0.6133ρ0

πa [kg/m⁴] (2.16)

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Figure 2.6: Speaker mounted in a vented-box enclosure

The acoustical impedance is easily converted to a mechanical equivalent where it is added to the mechanical impedance:

ZM =ZM0+S²_D(Zrad_r+Zrad_f) (2.17) Now the three parameters in (2.5) are written as, Mt which is the total mass of the diaphragm with assembly and air load, Rt which is the total resistance and Ct which is the total compliance of the suspension and air in box.

2.5.2 Vented box

The vented box loudspeaker is seen in figure 2.6. Here MAP is the mass of the air in the vent and SP is the cross-section of it.

Before the vented box circuit is written, the acoustic mass seen by the diaphragm must be converted to a mechanical equivalent and added to the mechanical mass. Both the acoustical masses MAB and MA1

are defined for the vented box loudspeaker the same way as for the closed box and can be added to the mechanical impedance in (2.7):

Z_M1=Z_M₀+S_D²(M_AB+M_A1)s (2.18)

The compliance and resistance, can not directly be converted into a mechanical equivalent as for the closed box system and thus a new acoustical impedance for the back of the diaphragm must be derived.

The total volume velocity emitted by the speaker U0is:

U₀(t) =U_D(t) +U_P(t) +U_L(t) =−UB(t) (2.19) where UD, UP and UL is the contribution from the diaphragm, port (vent) and losses, and UB is the volume velocity emitted into the box. The losses are due to the fact that the box is not ideal and some of the sound energy travels through the walls and air leaks. This is also the case for the closed box speaker, but in contrast to here, the effect with respect to the displacement is very small. The losses affects the displacement around the Helmholtz resonance.

Inserting the formulas in Laplace time for each volume velocity into (2.19) gives:

U0(s) =SDuD(s) + pc(s)

M_APs+pc(s)

R_AL =−pc(s)CABs (2.20)

By calculating the Laplace transform to this and rewriting it, the impedanceZrad_{V B} can be written:

Z_rad_{V B}(s) =− pc(s)

S_Du_D(s) =−pc(s)

U_D(s)= MAPRALs

M_APC_ABR_ALs²+M_APs+R_AL (2.21) The mass MAP and the complianceCAB forms a resonating circuit, with the resonance frequency, called the Helmholtz frequency, given by:

ω_B = 2πf_B= 1

√MAPCAB

(2.22)

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Here the diaphragm is short circuited and does not move (the loss causes it to move little), instead the vent and box resonate resulting a sound pressure. Furthermore, the quality factorQLcan be written as:

QL=RAL

rCAB

MAP

(2.23) The loss resistor RAL can not be calculated and can be very hard to measure. In [Leach, 1999] a thumb of rule is given. If the volume of the box is around 55l to 85l, thenQL= 7 is a good guess. If the volume is bigger thenQL must be less and vice versa.

The compliance is the same as for the closed box given in (2.14). The mass of the air in the vent is given by:

MAP = ρ0

SP

LP,ef f (2.24)

where LP,ef f is the effective length of the vent. The radiation impedance of a flanged tube with a radius ap is:

Za,r=ρ0c S_p

1−J1(2ωap/c)

ωa_p/c +jH1(2ωap/c) ωa_p/c

(2.25) where J1is the Bessel function and H1is the Struve function, both of first order; Their definition is given in appendix A.2. A good approximation is found for frequencies belowωap/c <0.5:

Za,r≈ρ0c S_p

1

2(ωap/c)²+j 8 3πωap/c

(2.26) The radiation impedance of an unflanged tube can be approximated as:

Z_a,r≈ ρ₀c Sp

1

4(ωa_p/c)²+ j0.61·ωa_p/c

(2.27) The last term in the brackets in both cases is the impedance of a acoustic mass corresponding to an extension in the length of the tube equal to 8a_p/3πand 0.61a_p. The typical use of a vent mounted in a box, one end is flanged and one is unflanged, gives the effective length of it:

L_{P,ef f} =L_P,phy+ ∆L_P =L_P,phy+ 8

3π + 0.61

a_p (2.28)

The magnitude and phase response for the true, the approximated and an acoustic mass respectively, is shown in figure 2.7. As will be shown in section 2.5.4, the vent only contributes to the sound pressure below about 200 Hz. Therefore the radiation impedance used throughout this thesis is the simple impedance of a acoustic mass.

Now the acoustic impedance for the vented box can be added to (2.18):

ZM =ZM1+S_D²Zrad_{V B}(s) (2.29)

Finally the transfer function for the volume velocity emitted from the vent can be written by combining (2.19) and (2.20):

UP(s)

U_D(s) = 1 CABMAPs²+^M_R^AP

AL + 1 (2.30)

2.5.3 Acoustic response

The on-axis sound pressure radiated by a circular piston (diaphragm) can be written as a point monopole source in free space if the shortest wavelength of sound considered is longer than any dimension of the loudspeaker:

pr(s) =ρ0sUD(s)e^−jkr

4πr (2.31)

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Figure 2.7: Acoustic radiation impedance and approximations to it

where UD =SDuD is the volume velocity, k= ω/cis the wave number and r is the distance from the diaphragm to the observation point. The complex exponential represents the phase delay caused by the propagation time delay from the diaphragm to the observation point. If the diaphragm is assumed to be mounted in a baffle (the longest wavelength of sound considered is shorter than any dimension of the loudspeaker), the term 4πrwill be replaced by 2πr.

The observation point is from now on considered to be 1m from the diaphragm, and then the pressure can be written as:

p_1m(s) = ρ₀

4πsU_D(s) (2.32)

For the vented box system the sum of the volume velocities from the diaphragm and vent must be used:

p1m(s) = ρ0

4πs(UD(s) +UP(s)) (2.33)

2.5.4 Linear frequency responds

If the voltage equation (2.2) and the force equation (2.6) is combined and the electrical impedance ZEB

and mechanical impedance ZM are used, then the transfer function for the velocity with the voltageu can be derived:

uD(s)

u(s) = Bl

Z_E(s)Z_M(s) +Bl² (2.34)

And given that UD=SDuD the pressure transfer function is nearly derived.

Closed box response

In the actual pressure response derived here, the electrical inductance is left out and given in a simple first order low-pass filter. This is often done, as it is the low frequencies there is of interest and the inductance only has a effect at high frequencies. With this R_E replacesZ_E in (2.34) and inserting it in (2.32) and rearranging it gives the pressure transfer function:

p_1m(s) = ρ₀ 4π

BlS_Du(s) REMt

(s/ω_C)²

(s/ωC)²+ (1/QT C)(s/ωC) + 1 (2.35)

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(a) (b)

Figure 2.8: SPL at 1m for a closed box loudspeaker

where ωC is the closed box system resonance frequency andQT S is its quality factor, given by:

ωC = 1

√MtCt

(2.36)

Q_{T C} = 1

Rt+Bl²/RE

rM_t Ct

(2.37) The low-pass properties of the driver due to the inductance, can be written as a simple first order low-pass filter:

T_u1(s) = 1 1 +s/ωu1

(2.38) where ωu1 is given by:

ω_u1= REMt

L_EM_D (2.39)

The sound pressure level and its phase is calculate and plotted in figure 2.8. The parameters for the loudspeaker constructed in section 2.7 has been used; the vent is blocked so the loudspeaker is assumed to be a closed box. As seen, the low frequency slope is 40dB/dec and the high frequency slope is 20dB/dec.

The upper cutoff frequency fu1 is at 1781Hz, which is much lower than shown in the datasheet. Actually it is because the model does not fit well at high frequencies due to eddy currents, see section 2.6.2 and diaphragm break up see section 3.1.2. In figure 2.9 the displacement response and its phase are plotted.

It can be seen that the displacement is a second order low-pass filter.

Vented box response

Again the electrical inductance is excluded and the pressure transfer function of a vented-box loudspeaker is given by:

p_1m(s) = ρ₀ 4π

BlS_Du(s) REMt

G_V(s) (2.40)

WhereGV(s) is the unity gain pressure transfer function and given by:

G_V(s) = (s/ω₀)⁴

(s/ω0)⁴+a3(s/ω0)³+a2(s/ω0)²+a1(s/ω0)¹+ 1 (2.41)

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(a) (b)

Figure 2.9: Displacement for a closed box loudspeaker, 0dB = 1mm

The coefficients are written as:

a1= 1 QL

√ h+

√ h QT S

a2= α+ 1

h +h+ 1 QLQT S

a3= 1 QT S

√ h+

√ h QL

(2.42) where QT S is the quality factor of the driver in free air, see (2.9) andαandhis given by:

α= VAS

V_AB h= fB

f_s (2.43)

In figure 2.10 the sound pressure levels and phases for the diaphragm, vent and their summation, is shown for a vented box system. As seen, at the Helmholtz resonance frequency, the diaphragm hardly moves and the sound pressure is radiated from the vent. Subtracting the vent response from the diaphragm response, gives the total pressure response. As they are in phase below the helmholtz frequency and in inverse phase at and above it, and as the vent is subtracted from the diaphragm, they are in phase atfB

and above, but out of phase below. The displacement function and its phase is seen in figure 2.11. As seen, the displacement is again a low-pass filter, but at fB it moves only a bit.

2.6 Extensions to the linear model

2.6.1 Temperature model

The variation of the temperature is relatively slow compared with the lowest frequency component used in loudspeakers, thus the electro-mechanical model is considered as a linear but time-variant system; see [Klippel, 2003].

The temperature of the voice-coil is important because firstly, high temperatures might damage the loudspeaker and secondly, changes in the temperature changes the electrical resistance:

R_E(T_a+ ∆T_v) =R_E(T_a)(1 +δ∆T_v) (2.44) whereTa is the ambient temperature and ∆Tv is the temperature difference between the ambient and the voice-coil. δis the conductivity whereδ= 0.0393K⁻¹for copper andδ= 0.0377K⁻¹for aluminium. The

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(a) (b)

Figure 2.10: SPL at 1m for a vented box loudspeaker

(a) (b)

Figure 2.11: Displacement for a vented box loudspeaker, 0dB = 1mm

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Figure 2.12: Linear thermal model

increased temperature causes power compression and lower efficiency/sensitivity, see??and??.

So far two models have been presented, the integrator based model by [Henricksen, 1987] and a model by Bang & Olufsen A/S [Chapman, 1998]. The former is the most common used in papers for documenta- tion, but has not been implemented in any products on the market. The latter is the only one that is implemented in a loudspeaker [Chapman, 2000]; the loudspeaker is the Bang & Olufsen Beolab 5. Despite that it does not compensate for the nonlinearities given in chapter 3, it compensates for the temperature changes and is actually the only loudspeaker on the market yet that compensates for unwanted properties of the speaker.

Integrator based model

A thermal model of the loudspeaker is seen in figure 2.12. The thermal model describes the relationship between the input power Pt dissipated into heat and voice-coil temperature Tv and is modeled with a third-order integrator. The first integrator represents the heating of the coil by using the thermal resistance Rtv and thermal capacity Ctv. The second integrator models the heating of the magnet using Rtm andCtm as thermal resistance and capacity and the third models the cabinet heating usingRtc and Ctc. ∆Tv represent the increase of the voice-coil temperature ∆Tv =Tv−Ta, ∆Tmis the increase of the magnet temperature and ∆T_c is the increase of the cabinet temperature. Often the third integrator is left out besides when using a small sealed enclosure, as suggested by [Behler and Bernhard, 1998].

The input power is given by:

Pt= u²

Z_min(T_v) (2.45)

whereZ_min(T_v) is the minimum impedance that is the DC resistance including additional resistance that will generate heat, which is due to eddy currents in the magnet structure; [Button, 1992].

When applying a stimulus with constant power to the loudspeaker the thermal system will go into a thermal equilibrium, given by:

∆Tvss= (Rtv+Rtm)Pt=RtPt (2.46)

for the voice-coil where ∆Tvssis the steady-state temperature. The steady-state magnet temperature is:

∆Tmss=RtmPt (2.47)

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Figure 2.13: Circuit of electrical part with eddy current loss

When switching on the input power, at time t =ts on, the observed variation of the temperature of the magnet varies exponentially with:

∆T_m(t) = ∆T_mss(1−e^−(t−t^{S on}^)/τ^m) (2.48) towards the steady-state temperature ∆T_mss. The time constant of the magnet structure is:

τ_m=R_tmC_tm (2.49)

After switching off the input power, at time t=t_{s of f}, the temperature difference between the voice-coil and magnet/frame is:

∆T_v(t)−∆T_m(t) = (∆T_vss−∆T_mss)e^−(t−t^{S of f}^)/τ^m (2.50) with the time constant:

τ_v=R_tvC_tv (2.51)

2.6.2 Eddy currents

Eddy currents in the iron pole structure causes the electrical impedance to behave differently from the normal series network of a resistance and a inductance. The effect was first described in [Thiele, 1961].

The effect is often modeled with a lossy inductor in the model, that is the inductor having a resistor in parallel [Leach, 1999]. This model can be used to fit the impedance as well as necessary over an adequate frequency range.

An extended version is often used, where a resistorR2 in parallel with an inductorL2 is put in series in the electrical network, as seen at figure 2.13, [Klippel, 2003]. See appendix B for these values of the test loudspeaker.

A third model is given in [Vanderkooy, 1989]. In here the inductance is said to be varying over frequency with:

Im{ZE} ∝p

f (2.52)

Vanderkooy shows that this model is better than the first one.

Notice, the eddy currents causesωu1 in (2.39) to change, increasing the upper cutoff frequency.

2.6.3 Frequency modulation (Doppler effect)

Doppler distortion is caused by the moving diaphragm. Short-wavelength components (high frequencies) are affected by resulting frequency modulation (doppler effect) of long-wavelength components, [Moir, 1974] and [Beers and Belar, 1981].

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Using the assumption of linear plane wave propagating, an equation is found for the square root of the ratio of the generated side band power to the total power, also known as distortion factor:

DF = s

1−

J0

2πˆx1

λˆ2

²

= 1

√2

2πxˆ1

λˆ2

(2.53) where J0 is the Bessel function of the first kind, ˆx1is the amplitude of the low frequency cone excursion and λ2 is the wavelength of the modulated frequency.

In [Klippel, 1992a] an algorithm for compensating the doppler distortion can be found.

2.7 Speaker construction

In the progress of this project a test loudspeaker has been developed. The linear and nonlinear (described in chapter 3) parameters of the loudspeaker are used in computer simulations to approach the real world.

And at last the control systems are applied on the loudspeaker to support the computer simulations.

The test loudspeaker is a small 2-way vented-box speaker with the following drivers²:

• Vifa TC14WG49-08 bass/mid-tone

• Peerless 53 NDT ’811435’ tweeter

The linear parameter for the drivers, measured with the Klippel analyzer system, are found in appendix [?]. Despite that only the bass/mid-tone speaker is going to linearized, a tweeter is included in the design for music playing; music without the high tones and only bass and mid, can be very tiring to listen too.

The tweeter is one of the cheapest from DST (41.90 Dkr + VAT, price at 10.000 pcs.) and therefore it suits well in the concept of a low price loudspeakers with electronic compensation, though it is not compensated here.

As well as the tweeter, also the bass/mid driver is a low cost one from DST (93.99 Dkr + VAT, price at 10.000 pcs.). Actually two bass/middle-tone loudspeakers have been bought; just in case if one is broken.

Cheaper drivers can be found made by other manufactures, but the supply of more precise information about the drivers than available at their homepage, and the possibility for personal correspondence, would be lost.

In this section the design of the test loudspeaker is presented. It is designed from the approach recom- mended in [Leach, 1999], which uses the Thiele-Small parameters:

• fs, Resonance frequency

• RE, DC resistance

• Q_{M S}, Mechanical quality factor

• Q_ES, Electrical quality factor

• Q_{T S}, Total quality factor

• VAS, Volume equivalent

The Thiele-Small parameters are measured for both of bass/mid drivers, but only the parameters from one of them is used to design the speaker. Two methods have been used for measuring the Thiele-Small parameters:

2See www.d-s-t.com

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(a) (b)

Figure 2.14: measured impedance and fitted impedance, (a) s1, (b) s2

• Approach given in [Leach, 1999], where the impedance is measured, and the parameters are found by analyzing the result and fitting of the impedance transfer function to it.

• Automatic measurements with ”Klippel Analyzer System”. The driver is connected to a system that measures and calculates the parameters automatically.

In order to evaluate the two results, the measured impedance were compared to the impedance calculated from the results. It was found that the result measured with the ”Klippel Analyzer System” were best.

The calculated and measured impedance for both drivers are seen at figure 2.14 and the belonging parameters are given in table 2.1.

At low frequencies the voice-coil inductance L_E is assumed to be short circuit and then the voice-coil impedance, as given in [Leach, 1999], can be written as:

ZV C(j2πf) =RE+RES

j(1/QM S)(f /fs)

1−(f /f_s)²+j(1/Q_{M S})(f /f_s) (2.54) whereRES =REQM S/QES. From the function it can be predicted that|ZV C(0)|=RE,|ZV C(j2πfs)|= RE+RES and|ZV C(j2πf)|=RE forf fs.

This equation excludes the voice-coil inductanceLE as it is assumed to be short circuited at low frequencies, where the Thiele-Small parameters are found. If this is the case the impedance function is flat at higher frequencies, and not increasing as in figure 2.14, where the inductance is included. The figure is automatically plotted by the Analyzer System software.

2.7.1 Loudspeaker alignment

The vented-box loudspeaker is designed by specifying an alignment. By defining a specific alignment, the form of the pressure response can be specified. The alignment is adapted by changing the volume of the

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SI s1 s2 tweeter

RE Ω 5.54 5.57 3.57

f_s Hz 50.1 45.8 1348

Q_{M S} N/A 2.77 2.49 2.94

Q_ES N/A 0.57 0.50 3.95

Q_{T S} N/A 0.47 0.41 1.68

V_AS l 10.6 12.99 -

Table 2.1: Linear parameters for the drivers used in the test loudspeaker

box and the dimension of the vent; length and cross area. Specifying the alignment is done in reference to the magnitude-squared function defined by |GV(jω)|². From (2.41) the magnitude-squared function can be found:

|GV(j2πf)|²= (f /f₀)⁸

(f /f0)⁸+A3(f /f0)⁶+A2(f /f0)⁴+A1(f /f0)²+ 1 (2.55) where theA coefficients are given by:

A₃=a²₃−2a₂ A₂= 2 +a²₂−2a₁a₃ A₁=a²₁−2a₂ (2.56) and wherea₁,a₂anda₃are given in 2.42. One of the three alignments that are commonly used, depending onQT S [Leach, 1999]:

• Butterworth B4 alignment (QT S= 0.4)

• Quasi-Butterworth QB3 alignment (QT S<0.4)

• Chebyshev C4 alignment (QT S >0.4)

As both drivers has a QT S above 0.4, the Chebyshev alignment is used. In this section the procedure for calculating the C4 alignment, derived in [Leach, 1999], is given and used.

But before starting to derive the alignment, some predefined values are set:

• The volume of the box must be 9l. This volume is chosen because a small speaker is wanted and given the measured values this seems reasonable.

• The vent chosen has a diameter equal to 4.3cm.

• Qlcan not be calculated, but it is needed in order to make the alignment. In [Leach, 1999] a rule of thumb is given. If the volume is between 55l and 85l a good choice would beQ_l= 7, if the volume is smaller it must be high and vice versa. Because the volume of this speaker is 9l,Q_l= 15.

The C4 alignment has a magnitude-squared function given by:

|GV(j2πf)|²= 1 +²

1 +²C₄²(f_n/f) (2.57)

whereis a parameter that specifies the amount of ripple andC4 is the fourth order Chebyshev polynomial given by:

C4(fn/f) = 8(fn/f)⁴−8(fn/f)²+ 1 (2.58) where fn is a normalization frequency the is related to the lower−3dB cutoff frequencyflby:

f_n= fl

√2

1 + q

1 + 4p

2 + 1/² ^1/2

(2.59)

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SI Vented-box

VAB l 9

S_P cm² 58.1

L_P cm 16.7

f_B Hz 49.25

f_l Hz 48.4

Q_l N/A 15

dB dB 0.000009≈0

k N/A 0.9477

Table 2.2: Calculated box parameters in alignment

The amount of ripple peak to peak in [dB] can be specified withdB andcan be calculated:

=p

10^dB^/10−1 (2.60)

Theacoefficients can be calculated by:

a1=kp 4 + 2√

2

D^1/4 a2= 1 +k² 1 +√ 2

D^1/2 a3= a1

D^1/2

1−1−k² 2√

2

(2.61) where kandD are given by:

k= tanh 1

4sinh⁻¹1

D= k⁴+ 6k²+ 1

8 (2.62)

If k= 1 then the alignment is equal to the Butterworth alignment and if k <1 it is a chebyshev. Then the following equation is solved for the positive real roots:

d⁴−A3d³−A2d²−A1d−1 = 0 (2.63) And again for the next equation, it is solved for the positive real roots:

r⁴−(a₃Q_L)r³−(a₁Q_L)r−1 = 0 (2.64) And with the solutions for d and r, the Helmholtz resonance frequency fB and the lower −3dB cutoff frequency can be calculated:

fB=r²fs fl=r

√

dfs (2.65)

In table 2.2 the resulting parameters for the C4 alignment is seen. With the predefined values a theoreti- cally alignment with no ripple is achieved. Given the value of k, it can be concluded that the alignment is close to be a B4 alignment.

At last, the length of the vent can be found by:

LP = c

2πfB

2

S_P VAB

−1.463 rS_P

π (2.66)

Where the last term is the end correction given in (2.28). The result is found in table 2.2.

2.7.2 Box construction and measurements

In figure 2.15 three drawings of the box, with inside dimensions, are seen, from the front, the side and the top where a cross section half way down is shown. As seen, the two sides, the back and front plate is angled so they are not in parallel with each other. This minimizes the possibility for standing waves

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Figure 2.15: Drawings of the loudspeaker box

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SI Vented-box

Vc l 9

S_P cm² 58.1

L_P cm 17

f_B Hz 51

f_l Hz 60

Table 2.3: Measured box parameters in alignment

inside the box. Standing waves occurs wherei·λ, where iis an integer, is equal to the distance between the two surfaces. They have a resonating effect that results in a deflection at the frequency. Only the top and bottom plate are in parallel.

To suppress standing waves even further, all inside walls, except the front, are lightly covered with fiber fill. The center of the box is not filled, as the air must able to travel between the bass/mid and the vent without resistance.

In order for the air to travel freely from inside of the box into the vent and back, good space from the back plate to the vent opening must be encountered. With the depth of the box in figure 2.15 and the calculated vent length, this is achieved.

The fourth and last drawing in figure 2.15 displays the positions of the bass/mid, vent and tweeter on the front plate. The outer black rectangle is the outside of the box, and the inner grey is the inside. The vent and the bass/mid are placed closely together. This is due to the fact that when measuring with a microphone, in front of the speaker, the radiated sound must sum up in the frequency range measured, and they only do if they are in phase. From that it can be concluded that the distance between the center of the bass/mid and the center of the vent, must be less than the length of half of the shortest wavelength measured.

All sides are made of 16mm MDF.

As a volume velocity is emitted from each of the bass/mid, vent and leaks in the box (2.19), the summed sound pressure can be difficult to measure outside of the box; they must all be measure individually and then added together. If the sound pressure instead is measured inside the box, only one measurement must be done as they are summed at low frequencies when the shortest wavelength of the sound is long compared to the inside dimensions (2.19). When measuring inside a box, the resulting pressure is somewhat different as if measure outside of the box. That is due to the fact that the impedance, for the sound radiated from the front of the diaphragm, is a mass sMA1 and the impedance for the sound radiated in back of the diaphragm is a compliance 1/sCAB. This affects the pressure response and in order to get the correct response, the measured response must be differentiated two times, or equivalent, multiplying the response with f².

In figure 2.16 a measurement of the sound pressure inside the box, after it is corrected, is seen. In table 2.3 the measured box parameters are seen.

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(a) (b)

Figure 2.16: Measurements of box alignment, (a) corrected sound pressure insight of box, (b) electrical impedance

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Compensation of Nonlinearities in Transducers